Question

1. consider the given right triangle. find the missing side lengths. x = select y = select show your work

Explanation:

Step1: Use trigonometric ratios

In a right - triangle, \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\) and \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Given \(\theta = 60^{\circ}\) and hypotenuse \(c = 12\).
For \(y\) (adjacent side to \(60^{\circ}\)): \(\cos60^{\circ}=\frac{y}{12}\). Since \(\cos60^{\circ}=\frac{1}{2}\), we have \(\frac{y}{12}=\frac{1}{2}\).

Step2: Solve for \(y\)

Cross - multiply: \(y = 12\times\frac{1}{2}=6\).
For \(x\) (opposite side to \(60^{\circ}\)): \(\sin60^{\circ}=\frac{x}{12}\). Since \(\sin60^{\circ}=\frac{\sqrt{3}}{2}\), we have \(\frac{x}{12}=\frac{\sqrt{3}}{2}\).

Step3: Solve for \(x\)

Cross - multiply: \(x = 12\times\frac{\sqrt{3}}{2}=6\sqrt{3}\).

Answer:

\(x = 6\sqrt{3}\), \(y = 6\)